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# Parallel Adaptive Wang Landau - GDR November 2011

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http://arxiv.org/abs/1109.3829

http://cran.r-project.org/web/packages/PAWL/index.html

http://statisfaction.wordpress.com/2011/09/21/density-exploration-and-wang-landau-algorithms-with-r-package/

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### Parallel Adaptive Wang Landau - GDR November 2011

1. 1. Wang–Landau algorithm Improvements 2D Ising model Conclusion Parallel Adaptive Wang–Landau Algorithm Pierre E. JacobCEREMADE - Universit´ Paris Dauphine & CREST, funded by AXA Research e 15 novembre 2011joint work with Luke Bornn (UBC), Arnaud Doucet (Oxford), Pierre Del Moral (INRIA & Universit´ de Bordeaux) e Pierre E. Jacob PAWL 1/ 18
2. 2. Wang–Landau algorithm Improvements 2D Ising model ConclusionOutline 1 Wang–Landau algorithm 2 Improvements Automatic Binning Parallel Interacting Chains Adaptive proposals 3 2D Ising model 4 Conclusion Pierre E. Jacob PAWL 2/ 18
3. 3. Wang–Landau algorithm Improvements 2D Ising model ConclusionWang–Landau Context unnormalized target density π on a state space X A kind of adaptive MCMC algorithm It iteratively generates a sequence Xt . The stationary distribution is not π itself. At each iteration a diﬀerent stationary distribution is targeted. Pierre E. Jacob PAWL 3/ 18
4. 4. Wang–Landau algorithm Improvements 2D Ising model ConclusionWang–Landau Partition the space The state space X is cut into d bins: d X = Xi and ∀i = j Xi ∩ Xj = ∅ i=1 Goal The generated sequence spends the same time in each bin Xi , within each bin Xi the sequence is asymptotically distributed according to the restriction of π to Xi . Pierre E. Jacob PAWL 4/ 18
5. 5. Wang–Landau algorithm Improvements 2D Ising model ConclusionWang–Landau Stationary distribution Deﬁne the mass of π over Xi by: ψi = π(x)dx Xi The stationary distribution of the WL algorithm is: 1 πψ (x) ∝ π(x) × ψJ(x) where J(x) is the index such that x ∈ XJ(x) Pierre E. Jacob PAWL 5/ 18
6. 6. Wang–Landau algorithm Improvements 2D Ising model ConclusionWang–Landau Example with a bimodal, univariate target density: π and two πψ corresponding to diﬀerent partitions. Original Density, with partition lines Biased by X Biased by Log Density 0 −2 −4 Log Density −6 −8 −10 −12 −5 0 5 10 15 −5 0 5 10 15 −5 0 5 10 15 X Pierre E. Jacob PAWL 6/ 18
7. 7. Wang–Landau algorithm Improvements 2D Ising model ConclusionWang–Landau Plugging estimates In practice we cannot compute ψi analytically. Instead we plug in estimates θt (i) of ψi at iteration t, and deﬁne the distribution πθt by: 1 πθt (x) ∝ π(x) × θt (J(x)) Metropolis–Hastings The algorithm does a Metropolis–Hastings step, aiming πθt at iteration t, generating a new point Xt . Pierre E. Jacob PAWL 7/ 18
8. 8. Wang–Landau algorithm Improvements 2D Ising model ConclusionWang–Landau Estimate of the bias The update of the estimated bias θt (i) is done according to: θt (i) ← θt−1 (i)[1 + γt (IXt ∈Xi − d −1 )] with γt a decreasing sequence or “step size”. E.g. γt = 1/t. Pierre E. Jacob PAWL 8/ 18
9. 9. Wang–Landau algorithm Improvements 2D Ising model ConclusionWang–Landau Result In the end we get: a sequence Xt asymptotically following πψ , as well as estimates θt (i) of ψi . Pierre E. Jacob PAWL 9/ 18
10. 10. Wang–Landau algorithm Automatic Binning Improvements Parallel Interacting Chains 2D Ising model Adaptive proposals ConclusionAutomate Binning Easily move from one bin to another Maintain some kind of uniformity within bins. If non-uniform, split the bin. Frequency Frequency Log density Log density (a) Before the split (b) After the split Pierre E. Jacob PAWL 10/ 18
11. 11. Wang–Landau algorithm Automatic Binning Improvements Parallel Interacting Chains 2D Ising model Adaptive proposals ConclusionParallel Interacting Chains (1) (N) N chains (Xt , . . . , Xt ) instead of one. targeting the same biased distribution πθt at iteration t, sharing the same estimated bias θt at iteration t. The update of the estimated bias becomes: N 1 θt (i) ← θt−1 (i)[1 + γt ( IX (j) ∈X − d −1 )] N t i j=1 Pierre E. Jacob PAWL 11/ 18
12. 12. Wang–Landau algorithm Automatic Binning Improvements Parallel Interacting Chains 2D Ising model Adaptive proposals ConclusionAdaptive proposals For continuous state spaces We can use the adaptive Random Walk proposal where the variance σt is learned along the iterations to target an acceptance rate. Robbins-Monro stochastic approximation update σt+1 = σt + ρt (2I(A > 0.234) − 1) Or alternatively Σt = δ × Cov (X1 , . . . , Xt ) Pierre E. Jacob PAWL 12/ 18
13. 13. Wang–Landau algorithm Improvements 2D Ising model Conclusion2D Ising model Higdon (1998), JASA 93(442) Target density Consider a 2D Ising model, with posterior density   π(x|y ) ∝ exp α I[yi = xi ] + β I[xi = xj ] i i∼j with α = 1, β = 0.7. The ﬁrst term (likelihood) encourages states x which are similar to the original image y . The second term (prior) favors states x for which neighbouring pixels are equal, like a Potts model. Pierre E. Jacob PAWL 13/ 18
14. 14. Wang–Landau algorithm Improvements 2D Ising model Conclusion2D Ising models (a) Original Image (b) Focused Region of Image Pierre E. Jacob PAWL 14/ 18
15. 15. Wang–Landau algorithm Improvements 2D Ising model Conclusion2D Ising models Iteration 300,000 Iteration 350,000 Iteration 400,000 Iteration 450,000 Iteration 500,000 40 Metropolis−Hastings 30 20 10 Pixel On X2 40 Off 30 Wang−Landau 20 10 10 20 30 40 10 20 30 40 10 20 30 40 10 20 30 40 10 20 30 40 X1 Figure: Spatial model example: states explored over 200,000 iterations for Metropolis-Hastings (top) and proposed algorithm (bottom). Pierre E. Jacob PAWL 15/ 18
16. 16. Wang–Landau algorithm Improvements 2D Ising model Conclusion2D Ising models Metropolis−Hastings Wang−Landau 40 30 Pixel 0.4 0.6 X2 20 0.8 1.0 10 10 20 30 40 10 20 30 40 X1 Figure: Spatial model example: average state explored with Metropolis-Hastings (left) and Wang-Landau after importance sampling (right). Pierre E. Jacob PAWL 16/ 18
17. 17. Wang–Landau algorithm Improvements 2D Ising model ConclusionConclusion Automatic binning We still have to deﬁne a range. Parallel Chains In practice it is more eﬃcient to use N chains for T iterations instead of 1 chain for N × T iterations. Adaptive Proposals Convergence results with ﬁxed proposals are already challenging, and making the proposal adaptive might add a layer of complexity. Pierre E. Jacob PAWL 17/ 18
18. 18. Wang–Landau algorithm Improvements 2D Ising model ConclusionBibliography Article: An Adaptive Interacting Wang-Landau Algorithm for Automatic Density Exploration, L. Bornn, P.E. Jacob, P. Del Moral, A. Doucet, available on arXiv. Software: PAWL, an R package, available on CRAN: install.packages("PAWL") References: F. Wang, D. Landau, Physical Review E, 64(5):56101 Y. Atchad´, J. Liu, Statistica Sinica, 20:209-233 e Pierre E. Jacob PAWL 18/ 18